In the above diagram x- axis is the real axis and y-axis is the imaginary axis. We will express real and imaginary part of the complex number in the form of r and θ where r is the modulus and θ is the argument of the complex number. θ is the angle made by the complex number with the real ( x) axis.

In the above diagram, we know that

r² = a² + b²

By the trigonometric ratios we know that

cos θ = a/r and sin θ = b/r.

we can rewrite it as

r cosθ = a and r sinθ = b.

Substituting the value of a and b in the rectangular coordinate form we get

z = r cosθ + i r sinθ

z = r(cosθ + i sinθ)

where r = ∣z∣ = √(a² + b²) and

θ= tan ⁻¹ (b/a)

Note: If a > 0 then θ= tan ⁻¹ (b/a).

If a < 0 then θ= tan ⁻¹ (b/a) + π

or

θ= tan ⁻¹ (b/a) + 180°

Example:

Express z= 1+i in the form of polar.

Solution:

Let us find the value of r.

In the given complex number a = 1 and b = 1

So the value of r is √(a² + b²)

r = √(1² + 1²)

= √2

and cos θ = a/r and sin θ = b/r.

cos θ = 1/√2 and sin θ = 1/√2

which implies θ = tan ⁻¹ (b/a)

= tan ⁻¹ (1/1).

= tan ⁻¹ (1).

= π/4.

So the polar form is z = 1/√2(cos π/4 + i sin π/4).

Example:

Convert in to rectangular form.

z = 2(cos 30° + i sin 30° )

Solution:

z = 2 cos30° + i 2sin 30°

Here

a = r cosθ and b = r sin θ.

So a = 2 cos 30° and b = 2 sin 30°

a = 2(0.87) and b = 2(0.5)

a = 1.74 and b = 1

The rectangular form is z = 1.74 + i

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